Finite Group Databasebeta

← #11739048

Subgroups — #11739048

Order 896.

438 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90416C211yesmaximal, characteristic, complemented
2#90416C211yesmaximal, characteristic, complemented
3C7×#487C211yesmaximal, characteristic, complemented
4#89860C211yesmaximal, characteristic, complemented
5#89857C211yesmaximal, characteristic, complemented
6#89591C214yesmaximal, complemented
7#89583C214yesmaximal, complemented
8#89467C211yesmaximal, characteristic, complemented
9#89457C211yesmaximal, characteristic, complemented
10C2×#1161—21no—
11C2×#1161C2212yes—
12C7×C4⋊Q8C2211yescharacteristic
13C7×C4⋊Q8C2211yescharacteristic
14#6510—21no—
15#6510C2212yes—
16C2×#1143C2214yes—
17C2×D56C2214yes—
18#6458—21no—
19#6458—21no—
20#6458—21no—
21#6458C2212yes—
22#6456—21no—
23#6456—21no—
24#6456—21no—
25#6456C2212yes—
26#6437C2211yescharacteristic
27#6437C2211yescharacteristic
28C42×D7C2211yescharacteristic
29C7×#103C2214yes—
30C4×C7×C8C2211yescharacteristic
31(C2×D7)⋊C8C2212yes—
32Dic7⋊C8C2212yes—
33#6383C2214yes—
34C4×C7⋊C8C2211yescharacteristic
35#3207—71nomaximal, Sylow
36C2×C7×Q8—21no—
37C2×C7×Q8C2312yes—
38#1161—44no—
39#1161—44no—
40C2×D28—22no—
41C2×D28—22no—
42C2×D28—21no—
43C2×D28—21no—
44C2×D28—21no—
45C2×D28—21no—
46C2×D28C2312yes—
47C2×D28C2312yes—
48C2×C4×D7—22no—
49C2×C4×D7—22no—
50C2×C4×D7D411yescharacteristic
51C2×C4×D7D411yescharacteristic
52C2×C4×D7C2311yescharacteristic
53C2×C7×C8C2312yes—
54C7×C4⋊C4—21no—
55C7×C4⋊C4—21no—
56C7×C4⋊C4—21no—
57C7×C4⋊C4C2312yes—
58C42×C7C2311yescharacteristic
59#1143—44no—
60(C2×D7)⋊C4—22no—
61(C2×D7)⋊C4—22no—
62(C2×D7)⋊C4—22no—
63(C2×D7)⋊C4—22no—
64C4×Dic7D411yescharacteristic
65C4×Dic7D411yescharacteristic
66C4×Dic7C2311yescharacteristic
67C2×C7⋊C8C2312yes—
68D56—44no—
69#532—71no—
70#532—71no—
71#493—71no—
72#492—71no—
73#487D711yescharacteristic, complemented
74#486—71no—
75#463—74no—
76#449—74no—
77#431—71no—
78C22×D7—22no—
79C22×D7—22no—
80C22×D7C2×D411yescharacteristic
81C7×Q8—24no—
82C7×Q8—42no—
83C2×C4×C7—22no—
84C2×C4×C7—22no—
85C2×C4×C7C2×D411yescharacteristic
86C2×C4×C7C2×D411yescharacteristic
87C2×C4×C7C2411yescharacteristic
88C2×Dic7C2×D411yescharacteristic
89C2×Dic7C2×D411yescharacteristic
90C2×Dic7C2×D411yescharacteristic
91D28—24no—
92D28—24no—
93D28—44no—
94D28—44no—
95D28—42no—
96D28—42no—
97D28—42no—
98D28—42no—
99C4×D7—22no—
100C4×D7—22no—
101C4×D7—22no—
102C4×D7—44no—
103C4×D7—44no—
104C7×C8—22no—
105C7⋊C8—22no—
106C2×#41—72no—
107C2×#41—141no—
108C2×SD16—74no—
109C2×D8—74no—
110C4⋊Q8C2×D711yescharacteristic
111C4⋊Q8C2×D711yescharacteristic
112#127—71no—
113#127—71no—
114#124—72no—
115#124—141no—
116#121—72no—
117#121—141no—
118#121—141no—
119#121—141no—
120C4×D4—72no—
121C4×D4—141no—
122C4×D4—141no—
123C4×D4—141no—
124C2×C42—71no—
125C4⋊C8—72no—
126#103C2×D714yes—
127#102—74no—
128C22⋊C8—72no—
129C4×C8—71no—
130C4×C8C2×D711yescharacteristic
131C22×C7C22×D411yescharacteristic
132C2×D7—21no—
133C2×D7—21no—
134C2×D7—21no—
135C2×D7—42no—
136C2×D7—42no—
137C2×D7—42no—
138C2×D7—42no—
139C2×D7—42no—
140C2×D7—42no—
141C4×C7—42no—
142C4×C7—42no—
143C4×C7#13512yes—
144C4×C7AGL(1,ℤ/8)12yes—
145C4×C7H(5,𝔽2)12yes—
146Dic7—21no—
147Dic7—21no—
148Dic7—21no—
149#41—284no—
150#41—284no—
151C2×Q8—21no—
152C2×Q8C22×D712yes—
153C2×D4—72no—
154C2×D4—72no—
155C2×D4—142no—
156C2×D4—142no—
157C2×D4—141no—
158C2×D4—141no—
159C2×D4—141no—
160C2×D4—141no—
161C22×C4—71no—
162C22×C4—71no—
163C22×C4—71no—
164C22×C4—142no—
165C22×C4—142no—
166SD16—284no—
167D8—284no—
168C2×C8—72no—
169C2×C8C22×D712yes—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4C22×D712yes—
174C22⋊C4—142no—
175C22⋊C4—142no—
176C22⋊C4—142no—
177C22⋊C4—142no—
178C42—71no—
179C42—71no—
180C42—71no—
181C42C22×D711yescharacteristic
182C2×C7#53511yescharacteristic
183C2×C7C2×#13511yescharacteristic
184C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
185D7—41no—
186D7—82no—
187D7—82no—
188C23—71no—
189C23—142no—
190C23—142no—
191Q8—24no—
192Q8—42no—
193D4—144no—
194D4—144no—
195D4—284no—
196D4—284no—
197D4—282no—
198D4—282no—
199D4—282no—
200D4—282no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—71no—
204C2×C4—71no—
205C2×C4—71no—
206C2×C4—142no—
207C2×C4—142no—
208C2×C4—142no—
209C2×C4—284no—
210C2×C4—284no—
211C2×C4D4×D711yescharacteristic
212C2×C4D4×D711yescharacteristic
213C2×C4C23×D711yescharacteristic
214C8—22no—
215C8—142no—
216C7#320711yescharacteristic, complemented, Sylow
217C22—141no—
218C22—141no—
219C22—141no—
220C22—282no—
221C22—282no—
222C22—282no—
223C22—282no—
224C22—282no—
225C22—282no—
226C22C2×D4×D711yescharacteristic
227C4—42no—
228C4—42no—
229C4—141no—
230C4—141no—
231C4—141no—
232C4#647712yes—
233C4#648212yes—
234C4#655312yes—
235C2—281no—
236C2—562no—
237C2—562no—
238C2#9036511yescharacteristic
239C2C2×#647711yescharacteristic
240C2C2×#648211yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#487C2
socleC22×C7C22×D4
hypercenter#487D7
hypocenterC7#3207
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #90416C2semidirect
2. #90416C2semidirect
3. C7×#487C2semidirect
4. #89860C2semidirect
5. #89857C2semidirect
6. #89591C2semidirect
7. #89583C2semidirect
8. #89467C2semidirect
9. #89457C2semidirect
73. #487D7semidirect
216. C7#3207semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#904162
C7×#4872
#898602
#898572
#895912
#895832
#894672
#894572
#32077